Definition:Maximal Element/Definition 2
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Definition
Let $\struct {S, \RR}$ be a relational structure.
Let $T \subseteq S$ be a subset of $S$.
An element $x \in T$ is a maximal element (under $\RR$) of $T$ if and only if:
- $\neg \exists y \in T: x \mathrel {\RR^\ne} y$
where $x \mathrel {\RR^\ne} y$ denotes that $x \mathrel \RR y$ but $x \ne y$.
Also defined as
Most treatments of the concept of a maximal element restrict the definition of the relation $\RR$ to the requirement that it be an ordering.
However, this is not strictly required, and this more general definition as used on $\mathsf{Pr} \infty \mathsf{fWiki}$ is of far more use.
Also see
Sources
- 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 14$: Order
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $23$. The Field of Rational Numbers
- 1967: Garrett Birkhoff: Lattice Theory (3rd ed.): $\S \text I.3$
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): maximal